Balls in 3D
Vincent Guillemot
11/11/2020
Definitions
Norm
A norm \(N : E \rightarrow \mathbb R\) is a function that is (\(\forall \mathbf{x}, \mathbf{y}\in E\), \(\forall \lambda \in \mathbb R\)) :
- positive, \(N(\mathbf{x}) \geq 0\),
- definite, \(N(\mathbf{x}) = 0 \Rightarrow \mathbf{x}= 0\),
- absolutely scalable, \(N(\lambda \mathbf{x}) \leq |\lambda|N(\mathbf{x})\),
- subadditive \(N(\mathbf{x}+\mathbf{y}) \leq N(\mathbf{x}) + N(\mathbf{y})\)
\(\ell_p\)-norm
An \(\ell_p\)-norm, denoted \(\|\cdot\|_p\), with \(p \in \mathbb N\) is defined as \[ \|\mathbf{x}\|_p = \left(\sum_{i=1}^n x_i^p\right)^{\frac{1}{p}} \]
\(\ell_p\)-ball
We define an \(\ell_p\)-ball of radius \(r > 0\) as \[ \mathscr{B}_p(r) = \left\{ \mathbf{x}\in E \text{ s.t. } \|\mathbf{x}\|_p \leq r\right\} \] In the following, we will consider that if \(r\) is not specified, it is equal to 1.
A detour by the 2D world
Let’s represent \(\ell_p\)-balls in 2D.
\(\mathscr{B}_1\)
\(\mathscr{B}_2\)
\(\mathscr{B}_\infty\)
